Iterated and mixed discriminants

A DICKENSTEIN

Conferencia; ZAG Seminar; 2022

Classical work by Salmon and Bromwichclassified singular intersections of two quadric surfaces. The basic idea ofthese results was already pursued by Cayley in connection with tangentintersections of conics in the plane and used by Schäfli for the study ofhyperdeterminants. More recently, the problem has been revisited with similartools in the context of geometric modeling and a generalization to the case oftwo higher dimensional quadric hypersurfaces was given by Ottaviani. In jointwork with Sandra di Rocco and Ralph Morrison, we propose and study ageneralization of this question for systems of Laurent polynomials with supporton a fixed point configuration.In the non-defective case, the closure of the locus ofcoefficients giving a non-degenerate multiple root of the system is defined bya polynomial called the mixed discriminant. We define a related polynomialcalled the multivariate iterated discriminant. This iterated discriminant iseasier to compute and we prove that it is always divisible by the mixeddiscriminant. We show that tangent intersections can be computed via iterationif and only if the singular locus of a corresponding dual variety hassufficiently high codimension. We also study when point configurationscorresponding to Segre-Veronese varieties and to the lattice points of planarsmooth polygons, have their iterated discriminant equal to their mixeddiscriminant